@inproceedings{Kelly1996closure,

  author    = {Wayne Kelly and

               William Pugh and

               Evan Rosser and

               Tatiana Shpeisman},

  title     = {Transitive Closure of Infinite Graphs and Its Applications},

  pages     = {126-140},

  editor    = {Chua-Huang Huang and

               P. Sadayappan and

               Utpal Banerjee and

               David Gelernter and

               Alexandru Nicolau and

               David A. Padua},

  booktitle = {Languages and Compilers for Parallel Computing, 8th International

               Workshop, LCPC'95, Columbus, Ohio, USA, August 10-12, 1995,

               Proceedings},

  publisher = {Springer},

  series    = {Lecture Notes in Computer Science},

  volume    = {1033},

  year      = {1996},

  isbn      = {3-540-60765-X},

}

@inproceedings{Beletska2009,

  author = {Beletska, Anna and Barthou, Denis and Bielecki, Wlodzimierz and Cohen, Albert},

  title = {Computing the Transitive Closure of a Union of Affine Integer Tuple Relations},

  booktitle = {COCOA '09: Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications},

  year = {2009},

  isbn = {978-3-642-02025-4},

  pages = {98--109},

  location = {Huangshan, China},

  doi = {10.1007/978-3-642-02026-1_9},

  publisher = {Springer-Verlag},

  address = {Berlin, Heidelberg},

}

@book{Schrijver1986,

    author  =	"Schrijver, Alexander",

    title   =	"Theory of Linear and Integer Programming",

    publisher	=   "John Wiley \& Sons",

    year    =	1986

}

@article{Tarjan1972,

    author = {Tarjan, Robert},

    journal = {SIAM Journal on Computing},

    number = {2},

    pages = {146--160},

    publisher = {SIAM},

    title = {Depth-First Search and Linear Graph Algorithms},

    volume = {1},

    year = {1972}

}

@TechReport{ Omega_calc,

    author = "Wayne Kelly and Vadim Maslov and William Pugh and Evan Rosser and Tatiana Shpeisman and Dave Wonnacott",

    title = "The {Omega} Calculator and Library",

    month = nov,

    institution = "University of Maryland",

    year = 1996

}

@TechReport{ Omega_lib,

    author = "Wayne Kelly and Vadim Maslov and William Pugh and Evan Rosser and Tatiana Shpeisman and Dave Wonnacott",

    title = "The {Omega} Library",

    month = nov,

    institution = "University of Maryland",

    year = 1996

}

@unpublished{Verdoolaege2009isl,

  author = "Verdoolaege, Sven",

  title = "An integer set library for program analysis",

  note = "Advances in the Theory of Integer Linear Optimization and its Extensions,AMS 2009 Spring Western Section Meeting, San Francisco, California, 25-26 April 2009",

  month = Apr,

  year = "2009",

  url = "https://lirias.kuleuven.be/handle/123456789/228373",

}

@article{Barthou2000MSE,

  author = {Barthou, Denis and Cohen, Albert and Collard, Jean-Fran\c{c}ois},

  title = {Maximal Static Expansion},

  journal = {Int. J. Parallel Program.},

  volume = {28},

  number = {3},

  year = {2000},

  issn = {0885-7458},

  pages = {213--243},

  doi = {10.1023/A:1007500431910},

  publisher = {Kluwer Academic Publishers},

  address = {Norwell, MA, USA},

}

@article{ Feautrier88parametric,

    author = "P. Feautrier",

    title = "Parametric Integer Programming",

    journal = "RAIRO Recherche Op\'erationnelle",

    volume = "22",

    number = "3",

    pages = "243--268",

    year = "1988",

}

@Article{ Fea91,

  author =       {Feautrier, P.},

  title =        {Dataflow analysis of array and scalar references},

  journal =      {International Journal of Parallel Programming},

  year =         {1991},

  OPTkey =         {},

  volume =       {20},

  number =       {1},

  OPTmonth =     {},

  pages =        {23--53},

  OPTnote =      {},

  OPTannote =    {},

}

@INPROCEEDINGS{BouletRe98,

  AUTHOR = {Pierre Boulet and Xavier Redon},

  TITLE = {Communication Pre-evaluation in {HPF}},

  BOOKTITLE = {EUROPAR'98},

  PAGES = {263--272},

  YEAR = 1998,

  VOLUME = 1470,

  series =	 {Lecture Notes in Computer Science},

  PUBLISHER = {Springer-Verlag, Berlin},

  ABSTRACT = {  Parallel computers are difficult to program efficiently.  We believe

  that a good way to help programmers write efficient programs is to

  provide them with tools that show them how their programs behave on

  a parallel computer.  Data distribution is the major performance

  factor of data-parallel programs and so automatic data layout for

  HPF programs has been studied by many researchers recently.  The

  communication volume induced by a data distribution is a good

  estimator of the efficiency of this data distribution.

  We present here a symbolic method to compute the communication

  volume generated by a given data distribution during the program

  writing phase (before compilation). We stay machine-independent to

  assure portability.  Our goal is to help the programmer understand

  the data movements its program generates and thus find a good data

  distribution. Our method is based on parametric polyhedral

  computations. It can be applied to a large class of regular codes.},

}

@INPROCEEDINGS {Verdoolaege2005experiences,

  AUTHOR = "Verdoolaege, Sven and Beyls, Kristof and Bruynooghe, Maurice and Catthoor, Francky",

  TITLE = {{E}xperiences with enumeration of integer projections of parametric polytopes},

  BOOKTITLE = {{P}roceedings of 14th {I}nternational {C}onference on {C}ompiler {C}onstruction, {E}dinburgh, {S}cotland},

  YEAR = {2005},

  EDITOR = {Bodik, R.},

  VOLUME = 3443,

    pages = "91-105",

    series = "Lecture Notes in Computer Science",

    publisher = "Springer-Verlag",

    address = "Berlin",

    doi = "10.1007/b107108",

}

@article{Detlefs2005simplify,

 author = {David Detlefs and Greg Nelson and James B. Saxe},

 title = {Simplify: a theorem prover for program checking},

 journal = {J. ACM},

 volume = {52},

 number = {3},

 year = {2005},

 issn = {0004-5411},

 pages = {365--473},

 doi = {10.1145/1066100.1066102},

 publisher = {ACM},

 address = {New York, NY, USA},

 }

@phdthesis{Nelson1980phd,

 author = {Charles Gregory Nelson},

 title = {Techniques for program verification},

 year = {1980},

 order_no = {AAI8011683},

 school = {Stanford University},

 address = {Stanford, CA, USA},

 }

@article{Woods2003short,

    year = 2003,

    Journal = "J. Amer. Math. Soc.",

    volume =  16,

    pages = "957--979",

    month = apr,

    title = {{Short rational generating functions for lattice point

        problems}},

    author = {Alexander Barvinok and Kevin Woods},

}

@misc{barvinok-0.22,

  author = {Sven Verdoolaege},

  title = {{\texttt{barvinok}}, version 0.22},

  howpublished = {Available from \url{http://freshmeat.net/projects/barvinok/}},

  year = 2006

}

@inproceedings{DeLoera2004Three,

    title = "Three Kinds of Integer Programming Algorithms based on Barvinok's Rational Functions",

    author = "De Loera, J. A. and D. Haws and R. Hemmecke and P. Huggins and R. Yoshida",

    booktitle = "Integer Programming and Combinatorial Optimization: 10th International IPCO Conference",

    year = "2004",

    month = jan,

    series = "Lecture Notes in Computer Science",

    Volume = 3064,

    Pages = "244-255",

}

@TechReport{Feautrier02,

  author = 	 {P. Feautrier and J. Collard and C. Bastoul},

  title = 	 {Solving systems of affine (in)equalities},

  institution =  {PRiSM, Versailles University},

  year = 	 2002

}

@article{ Feautrier92multi,

    author = "Paul Feautrier",

    title = "Some Efficient Solutions to the Affine Scheduling Problem. {P}art {II}. Multidimensional Time",

    journal = "International Journal of Parallel Programming",

    volume = "21",

    number = "6",

    pages = "389--420",

    year = "1992",

    month = dec,

    url = "citeseer.nj.nec.com/article/feautrier92some.html",

}

@misc{Bygde2010licentiate,

   author = {Stefan Bygde},

   title = {Static {WCET} Analysis based on Abstract Interpretation and Counting of Elements},

   month = {March},

   year = {2010},

   howpublished = {Licentiate thesis},

   publisher = {M{\"{a}}lardalen University Press},

   url = {http://www.mrtc.mdh.se/index.php?choice=publications&id=2144},

}

@phdthesis{Meister2004PhD,

	title = {Stating and Manipulating Periodicity in the Polytope Model. Applications to Program Analysis and Optimization},

	author= {Beno\^it Meister},

 	school = {Universit\'e Louis Pasteur},

	month = Dec,

	year  = {2004},

}

@inproceedings{Meister2008,

  author = {Beno\^it Meister and Sven Verdoolaege},

  title = {Polynomial Approximations in the Polytope Model: Bringing the Power

  of Quasi-Polynomials to the Masses},

  year = {2008},

  booktitle = {Digest of the 6th Workshop on Optimization for DSP and Embedded Systems, ODES-6},

  editor = "Jagadeesh Sankaran and Vander Aa, Tom",

  month = apr,

}

@misc{Galea2009personal,

    author = "Fran\c{c}ois Galea",

    title = "personal communication",

    year = 2009,

    month = nov,

}

@misc{PPL,

  author = "R. Bagnara and P. M. Hill and E. Zaffanella",

  title = "The {Parma Polyhedra Library}",

  howpublished = {\url{http://www.cs.unipr.it/ppl/}},

}

@TECHREPORT{Cook1991implementation,

AUTHOR={William Cook and Thomas Rutherford and Herbert E. Scarf and David F. Shallcross},

TITLE={An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming},

YEAR=1991,

MONTH=Aug,

INSTITUTION={Cowles Foundation, Yale University},

TYPE={Cowles Foundation Discussion Papers},

NOTE={available at \url{http://ideas.repec.org/p/cwl/cwldpp/990.html}},

NUMBER={990},

}

 @article{Karr1976affine,

author={ Michael Karr},

title={ Affine Relationships Among Variables of a Program },

journal={Acta Informatica},

Volume={6},

pages={133-151},

year={1976},

publisher={Springer-Verlag},

ignore={ },

}

@PhdThesis{Verhaegh1995PhD,

	title = "Multidimensional Periodic Scheduling",

	author = "Wim F. J. Verhaegh",

	school = "Technische Universiteit Eindhoven",

	year = 1995,

}

@INPROCEEDINGS{Seghir2006minimizing,

  AUTHOR = "Rachid Seghir and Vincent Loechner",

  TITLE = {Memory Optimization by Counting Points in Integer Transformations of Parametric Polytopes},

  BOOKTITLE = {{P}roceedings of the {I}nternational {C}onference on {C}ompilers, {A}rchitectures, and {S}ynthesis for {E}mbedded Systems, CASES 2006, {S}eoul, {K}orea},

  month = oct,

  YEAR = {2006}

}

@misc{DeSmet2010personal,

    author = "De Smet, Sven",

    title = "personal communication",

    year = 2010,

    month = apr,

}